okay, here's what i recommend:
back up your data first.
run error-checking on the drive on the machine that can read it, and see if it fixes any issues. Right click on the F: under my computer, click properties, and then go to the "Tools" tab, i believe it is, and run error-checking. make sure both boxes are checked before hitting start.
this will take some time. when it's done, plug it into the other machine. if it still doesn't read, you can try running the same error-check on that machine. also, do you have any kind of special security on compression on your external drive that would prevent it from being read on other computers?
also, check other USB ports and make sure other USB devices are functioning.
let me know what you find.